Solved Problems on Limits and Continuity

1. Solved Problems on Limits and Continuity

2. Overview of Problems 2 1 4 3 5 6 7 8 9 10 Mika Seppälä: Limits and Continuity

3. Overview of Problems 11 12 13 14 15 Mika Seppälä: Limits and Continuity

4. Main Methods of Limit Computations The following undefined quantities cause problems: 1 In the evaluation of expressions, use the rules 2 If the function, for which the limit needs to be computed, is defined by an algebraic expression, which takes a finite value at the limit point, then this finite value is the limit value. 3 If the function, for which the limit needs to be computed, cannot be evaluated at the limit point (i.e. the value is an undefined expression like in (1)), then find a rewriting of the function to a form which can be evaluated at the limit point. 4 Mika Seppälä: Limits and Continuity

5. 1 Frequently needed rule If a square root appears in the expression, then multiply and divide by the conjugate of the square root expression. 3 4 Use the fact that Main Computation Methods Cancel out common factors of rational functions. 2 Mika Seppälä: Limits and Continuity

6. Continuity of Functions Functions defined by algebraic or elementary expressions involving polynomials, rational functions, trigonometric functions, exponential functions or their inverses are continuous at points where they take a finite well defined value. 1 A function f is continuous at a point x = a if 2 Used to show that equations have solutions. The following are not continuous x =0: 3 4 Intermediate Value Theorem for Continuous Functions If f is continuous, f(a) < 0 and f(b) > 0, then there is a point c between a and b so that f(c) = 0. Mika Seppälä: Limits and Continuity

7. Limits by Rewriting Problem 1 Solution Mika Seppälä: Limits and Continuity

8. Limits by Rewriting Problem 2 Solution Mika Seppälä: Limits and Continuity

9. Limits by Rewriting Problem 3 Solution Rewrite Mika Seppälä: Limits and Continuity

10. Limits by Rewriting Problem 4 Solution Rewrite Next divide by x. Mika Seppälä: Limits and Continuity

11. Limits by Rewriting Problem 5 Solution Rewrite Next divide by x. Mika Seppälä: Limits and Continuity

12. Limits by Rewriting Problem 6 Solution Mika Seppälä: Limits and Continuity

13. Limits by Rewriting Problem 7 Rewrite: Solution Mika Seppälä: Limits and Continuity

14. Limits by Rewriting Problem 8 Rewrite: Solution Mika Seppälä: Limits and Continuity

15. Limits by Rewriting Problem 9 Solution Rewrite Mika Seppälä: Limits and Continuity

16. Here we used the fact that all sin(x)/x terms approach 1 as x  0. Limits by Rewriting Problem 9 Solution(cont’d) Rewrite Next divide by x. Mika Seppälä: Limits and Continuity

17. One-sided Limits Problem 10 Solution Mika Seppälä: Limits and Continuity

18. Continuity Problem 11 Solution Mika Seppälä: Limits and Continuity

19. Continuity Problem 12 Solution Mika Seppälä: Limits and Continuity

20. Continuity Problem 13 Solution Mika Seppälä: Limits and Continuity

21. Answer Removable Not removable Removable Continuity Problem 14 Mika Seppälä: Limits and Continuity

22. Continuity Problem 15 Solution By the intermediate Value Theorem, a continuous function takes any value between any two of its values. I.e. it suffices to show that the function f changes its sign infinitely often. Mika Seppälä: Limits and Continuity